Lighthouse

The man, 180 cm tall, walks along the seafront directly to the lighthouse. The male shadow caused by the beacon light is initially 5.4 meters long. When the man approaches the lighthouse by 90 meters, its shadow is shorter by 3 meters. How tall is the lighthouse, and how far is the man away from it?

Correct answer:

a =  55.44 m
b =  160.92 m

Step-by-step explanation:

$$\begin{gathered} m=180\text{cm}\to \text{m}=180:100\text{m}=1.8\text{m} \\ t=5.4\text{m} \\ x=90\text{m} \\ y=3\text{m} \\ a:(b+t)=m:t \\ a:(b-x+y)=m:(t-y) \\ a=m/t\cdot (b+t) \\ a=m/(t-y)\cdot (b-x+y) \\ a=1.8/5.4\cdot (b+5.4) \\ a=1.8/(5.4-3)\cdot (b-90+3) \\ a-0.333333b=1.8 \\ a-0.75b=-65.25 \\ Row2-Row1\to Row2 \\ a-0.33b=1.8 \\ -0.42b=-67.05 \\ b=\frac{-67.05}{-0.41666667}=160.92 \\ a=1.8+0.33333333333333b=1.8+0.33333333\cdot 160.92=55.44 \\ a=\frac{1386}{25}=55.44 \\ b=\frac{4023}{25}=160.92 \end{gathered}$$

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